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(b)   Deduce from (a) that, if we put x' --

continuous involution on A such that (#')* > (#*)' for some x e A.

12.    Let E be a Hilbert space. Show that a self-adjoint compact operator U is a Hilbert-
Schmidt operator if and only if, (Art) being the sequence of eigenvalues of U eacl
counted according to its multiplicity, the sum]T |An|2 is finite. The sum is then equa

to ||ffHI.

13.    (a)   The norm ||||2 is defined as in (15.4.8) when E is a finite-dimensional Hilber
space. In a Hilbert space of dimension n, give an example of an operator it such thai

<       1           2

(b) Let E be a Hilbert space of infinite dimension. Give an example of a se-
quence (un) of Hilbert-Schmidt operators on E such that the sequence of numbers
l|w?ir||2/WI! tends to 0 as n tends to oo.

14.    Let X be a (metrizable, separable) locally compact space, and let ft be a positive
measure on X. Let K(JC, y) be a function on X x X, belonging to &c$i x X, ft  ft).

For each function /e &c(X, ft), the function x\-*\ K(x,y)f(y) d^(y) is defined al-
most everywhere, and its class belongs to L(X, ft) (use the Lebesgue-Fubini theorem
and the Cauchy-Schwarz inequality). If this class is denoted by U / show that U is
a Hilbert-Schmidt operator; U is said to be associated to the kernel K. Show that
II U\\2 = N2(K) and that U* is associated to the kernel (x, y)^K(y,x). If Ul9 U2 are
associated to kernels KI, K2 respectively, belonging to ^c(X x X, ft (x) ft), then
Ui U2 is associated to the kernel

K(*^) =

15.    Let E be an infinite-dimensional Hilbert space and U a Hilbert-Schmidt operator on
E. If A is a regular value for /, show that the operator (U A  IE)"* is of the form
V A"1 * 1E> where Kis a Hilbert-Schmidt operator. Deduce that the spectrum of
U in the algebra -Sf(E) is the same as in the algebra obtained by adjoining a unit
element to &2(E). For every function / which is analytic in an open set containing
Sp(7), the operator f(U) (Section 15.2, Problem 11) is a Hilbert-Schmidt operator.

16.    Let HI, ..., Hn be self-adjoint operators, each pair of which commute, on a separable
Hilbert space E, and suppose that \\ffj\( ^ 1 and that \\HjHk\\ ^IJ~"1 for ally, k,
where e is a real number such that 0 g e < 1. Then

||#i +    + -ft|| ^ (1 + \/fi)/(l - e)

(" Collar's lemma ").

(Using the Gelfand-Neumark theorem, reduce to proving the same property for
real numbers ult.,., un. For each t i> 0, let v(t) be the number of indices/ such that
\uj\>t. Remark that

and prove that v(t) :g 1 + [2 log //log e], by observing that if \uj\et and \uk\ iy = 0,